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Properties

Label 38.7.d.a
Level $38$
Weight $7$
Character orbit 38.d
Analytic conductor $8.742$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,7,Mod(27,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.27");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.74205517755\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9440 x^{18} + 37579488 x^{16} + 83109028728 x^{14} + 112838424559344 x^{12} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{4}\cdot 19^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (\beta_{6} + \beta_{2} - 2) q^{3} + ( - 32 \beta_{2} + 32) q^{4} + (\beta_{12} - \beta_{6} + \cdots + 11 \beta_{2}) q^{5}+ \cdots + (\beta_{15} + 2 \beta_{14} + \cdots + 221) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + (\beta_{6} + \beta_{2} - 2) q^{3} + ( - 32 \beta_{2} + 32) q^{4} + (\beta_{12} - \beta_{6} + \cdots + 11 \beta_{2}) q^{5}+ \cdots + (1111 \beta_{19} + 653 \beta_{18} + \cdots + 62150) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} + 320 q^{4} + 112 q^{5} + 80 q^{6} - 208 q^{7} + 2200 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} + 320 q^{4} + 112 q^{5} + 80 q^{6} - 208 q^{7} + 2200 q^{9} - 284 q^{11} + 10500 q^{13} + 1248 q^{14} + 14136 q^{15} - 10240 q^{16} - 11684 q^{17} + 12862 q^{19} + 7168 q^{20} + 2916 q^{21} + 52704 q^{22} + 8488 q^{23} - 2560 q^{24} - 63842 q^{25} - 43968 q^{26} - 3328 q^{28} + 137760 q^{29} - 94688 q^{30} - 250194 q^{33} + 70560 q^{34} - 6916 q^{35} - 70400 q^{36} - 77232 q^{38} + 9672 q^{39} + 109206 q^{41} - 92672 q^{42} + 35572 q^{43} - 4544 q^{44} + 131264 q^{45} - 361184 q^{47} + 30720 q^{48} + 259740 q^{49} - 496512 q^{51} + 336000 q^{52} - 236172 q^{53} + 375728 q^{54} + 56760 q^{55} + 314796 q^{57} - 225600 q^{58} + 1310610 q^{59} + 452352 q^{60} + 83552 q^{61} + 225792 q^{62} + 553364 q^{63} - 655360 q^{64} - 4736 q^{66} + 806646 q^{67} - 747776 q^{68} - 245664 q^{70} - 869220 q^{71} + 353280 q^{72} - 207422 q^{73} + 1460832 q^{74} - 140096 q^{76} - 3988336 q^{77} - 85008 q^{78} - 1706808 q^{79} + 114688 q^{80} + 2303170 q^{81} + 887712 q^{82} + 3527548 q^{83} - 5604 q^{85} + 195792 q^{86} - 8414832 q^{87} + 708432 q^{89} - 4483104 q^{90} - 1914384 q^{91} - 271616 q^{92} + 3537876 q^{93} - 2820356 q^{95} - 163840 q^{96} + 5113242 q^{97} - 612480 q^{98} + 603704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 9440 x^{18} + 37579488 x^{16} + 83109028728 x^{14} + 112838424559344 x^{12} + \cdots + 83\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!81 \nu^{18} + \cdots + 28\!\cdots\!48 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!89 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92\!\cdots\!83 \nu^{19} + \cdots + 83\!\cdots\!88 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!11 \nu^{19} + \cdots - 19\!\cdots\!56 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{19} + \cdots + 11\!\cdots\!92 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{19} + \cdots - 11\!\cdots\!92 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{19} + \cdots + 24\!\cdots\!36 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!11 \nu^{19} + \cdots + 24\!\cdots\!36 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92\!\cdots\!83 \nu^{19} + \cdots - 65\!\cdots\!20 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!95 \nu^{19} + \cdots - 22\!\cdots\!48 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 45\!\cdots\!77 \nu^{19} + \cdots - 99\!\cdots\!32 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!97 \nu^{19} + \cdots - 69\!\cdots\!76 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!97 \nu^{19} + \cdots - 25\!\cdots\!96 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21\!\cdots\!65 \nu^{19} + \cdots - 11\!\cdots\!16 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!65 \nu^{19} + \cdots + 11\!\cdots\!16 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!73 \nu^{19} + \cdots - 41\!\cdots\!08 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 21\!\cdots\!73 \nu^{19} + \cdots + 41\!\cdots\!08 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 41\!\cdots\!87 \nu^{19} + \cdots - 13\!\cdots\!72 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 41\!\cdots\!53 \nu^{19} + \cdots + 13\!\cdots\!72 ) / 94\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} - 4\beta_{6} - 4\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{14} - 24 \beta_{8} - 24 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - \beta_{4} + \cdots - 1890 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 162 \beta_{19} - 162 \beta_{18} - 45 \beta_{17} - 45 \beta_{16} + 31 \beta_{15} + 31 \beta_{14} + \cdots + 20350 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 394 \beta_{19} + 394 \beta_{18} - 222 \beta_{17} + 222 \beta_{16} - 2168 \beta_{15} + 2168 \beta_{14} + \cdots + 1397350 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 224538 \beta_{19} + 224538 \beta_{18} + 72264 \beta_{17} + 72264 \beta_{16} - 110036 \beta_{15} + \cdots - 31101960 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1734108 \beta_{19} - 1734108 \beta_{18} + 782295 \beta_{17} - 782295 \beta_{16} + 4558569 \beta_{15} + \cdots - 2634390176 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 269118981 \beta_{19} - 269118981 \beta_{18} - 96766590 \beta_{17} - 96766590 \beta_{16} + \cdots + 43692674682 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5547034968 \beta_{19} + 5547034968 \beta_{18} - 2279633436 \beta_{17} + 2279633436 \beta_{16} + \cdots + 5721074436072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 641880228630 \beta_{19} + 641880228630 \beta_{18} + 248109227673 \beta_{17} + 248109227673 \beta_{16} + \cdots - 119859149364038 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15896064313532 \beta_{19} - 15896064313532 \beta_{18} + 6271094686164 \beta_{17} - 6271094686164 \beta_{16} + \cdots - 13\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15\!\cdots\!80 \beta_{19} + \cdots + 32\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 43\!\cdots\!32 \beta_{19} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 38\!\cdots\!00 \beta_{19} + \cdots - 85\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11\!\cdots\!84 \beta_{19} + \cdots - 83\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 97\!\cdots\!32 \beta_{19} + \cdots + 22\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 30\!\cdots\!32 \beta_{19} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 24\!\cdots\!48 \beta_{19} + \cdots - 59\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 80\!\cdots\!28 \beta_{19} + \cdots - 54\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 63\!\cdots\!28 \beta_{19} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(1 - \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
33.8996i
22.2621i
19.8056i
− 25.0158i
− 50.9515i
40.1546i
19.6100i
5.69488i
− 30.5778i
− 34.8817i
− 33.8996i
− 22.2621i
− 19.8056i
25.0158i
50.9515i
− 40.1546i
− 19.6100i
− 5.69488i
30.5778i
34.8817i
−4.89898 − 2.82843i −32.0827 − 18.5229i 16.0000 + 27.7128i −25.9363 + 44.9230i 104.782 + 181.487i −644.489 − 181.019i 321.699 + 557.199i 254.123 − 146.718i
27.2 −4.89898 − 2.82843i −22.0043 − 12.7042i 16.0000 + 27.7128i −85.1965 + 147.565i 71.8656 + 124.475i 473.710 − 181.019i −41.7080 − 72.2405i 834.752 − 481.944i
27.3 −4.89898 − 2.82843i −19.8769 − 11.4759i 16.0000 + 27.7128i 99.9842 − 173.178i 64.9176 + 112.441i 131.431 − 181.019i −101.107 − 175.122i −979.641 + 565.596i
27.4 −4.89898 − 2.82843i 18.9396 + 10.9348i 16.0000 + 27.7128i −55.1543 + 95.5300i −61.8564 − 107.138i −148.805 − 181.019i −125.361 − 217.132i 540.399 − 312.000i
27.5 −4.89898 − 2.82843i 41.4005 + 23.9026i 16.0000 + 27.7128i 94.3029 − 163.337i −135.214 − 234.197i 72.4666 − 181.019i 778.168 + 1347.83i −923.976 + 533.458i
27.6 4.89898 + 2.82843i −35.0502 − 20.2362i 16.0000 + 27.7128i −32.0278 + 55.4737i −114.473 − 198.274i 487.742 181.019i 454.510 + 787.234i −313.807 + 181.176i
27.7 4.89898 + 2.82843i −17.2580 − 9.96392i 16.0000 + 27.7128i −17.0310 + 29.4985i −56.3644 − 97.6261i −265.741 181.019i −165.941 − 287.418i −166.869 + 96.3417i
27.8 4.89898 + 2.82843i −5.20717 − 3.00636i 16.0000 + 27.7128i 116.624 − 201.999i −17.0065 − 29.4562i −326.637 181.019i −346.424 − 600.023i 1142.68 − 659.725i
27.9 4.89898 + 2.82843i 26.2059 + 15.1300i 16.0000 + 27.7128i −88.5587 + 153.388i 85.5881 + 148.243i −258.032 181.019i 93.3327 + 161.657i −867.695 + 500.964i
27.10 4.89898 + 2.82843i 29.9332 + 17.2819i 16.0000 + 27.7128i 48.9935 − 84.8592i 97.7614 + 169.328i 374.355 181.019i 232.830 + 403.274i 480.036 − 277.149i
31.1 −4.89898 + 2.82843i −32.0827 + 18.5229i 16.0000 − 27.7128i −25.9363 − 44.9230i 104.782 − 181.487i −644.489 181.019i 321.699 − 557.199i 254.123 + 146.718i
31.2 −4.89898 + 2.82843i −22.0043 + 12.7042i 16.0000 − 27.7128i −85.1965 − 147.565i 71.8656 − 124.475i 473.710 181.019i −41.7080 + 72.2405i 834.752 + 481.944i
31.3 −4.89898 + 2.82843i −19.8769 + 11.4759i 16.0000 − 27.7128i 99.9842 + 173.178i 64.9176 − 112.441i 131.431 181.019i −101.107 + 175.122i −979.641 − 565.596i
31.4 −4.89898 + 2.82843i 18.9396 − 10.9348i 16.0000 − 27.7128i −55.1543 − 95.5300i −61.8564 + 107.138i −148.805 181.019i −125.361 + 217.132i 540.399 + 312.000i
31.5 −4.89898 + 2.82843i 41.4005 − 23.9026i 16.0000 − 27.7128i 94.3029 + 163.337i −135.214 + 234.197i 72.4666 181.019i 778.168 − 1347.83i −923.976 − 533.458i
31.6 4.89898 − 2.82843i −35.0502 + 20.2362i 16.0000 − 27.7128i −32.0278 − 55.4737i −114.473 + 198.274i 487.742 − 181.019i 454.510 − 787.234i −313.807 − 181.176i
31.7 4.89898 − 2.82843i −17.2580 + 9.96392i 16.0000 − 27.7128i −17.0310 − 29.4985i −56.3644 + 97.6261i −265.741 − 181.019i −165.941 + 287.418i −166.869 − 96.3417i
31.8 4.89898 − 2.82843i −5.20717 + 3.00636i 16.0000 − 27.7128i 116.624 + 201.999i −17.0065 + 29.4562i −326.637 − 181.019i −346.424 + 600.023i 1142.68 + 659.725i
31.9 4.89898 − 2.82843i 26.2059 − 15.1300i 16.0000 − 27.7128i −88.5587 − 153.388i 85.5881 − 148.243i −258.032 − 181.019i 93.3327 − 161.657i −867.695 − 500.964i
31.10 4.89898 − 2.82843i 29.9332 − 17.2819i 16.0000 − 27.7128i 48.9935 + 84.8592i 97.7614 − 169.328i 374.355 − 181.019i 232.830 − 403.274i 480.036 + 277.149i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.7.d.a ✓ 20
3.b odd 2 1 342.7.m.a 20
19.d odd 6 1 inner 38.7.d.a ✓ 20
57.f even 6 1 342.7.m.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.d.a ✓ 20 1.a even 1 1 trivial
38.7.d.a ✓ 20 19.d odd 6 1 inner
342.7.m.a 20 3.b odd 2 1
342.7.m.a 20 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 71\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 73\!\cdots\!49 \) Copy content Toggle raw display
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